Find ?n(A?B').

Question

Let A and B be subsets of U with ?n(A)=49, ?n(B)=64, ?n(A')=37, and ?n(A∩B)=30.
Find ?n(A∪B').

Answer 1

n(A)=49

n(B)=64

n(A')=37

n(A∩B)=30

because of the intersection being 30, there are 49-30=19 elements in A only

because of the intersection being 30, there are 64-30=34 elements in B only

if n(A')=37 and B has 34 elements only in B, there must be 3 other elements in the Universe not in A or in B

n(B')=19 elements from A plus the 3 elements in the Universe not in A or in B

n(AUB')=19+3=22 because we don't count the 19 elements in A twice